Portfolios that Contain Risky Assets 12 Growth Rate Mean and Variance Estimators

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1 Portfolios that Contain Risky Assets 12 Growth Rate Mean and Variance Estimators C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling April 11, 2017 version c 2017 Charles David Levermore

2 Portfolios that Contain Risky Assets Part II: Stochastic Models 11. Independent, Identically-Distributed Models 12. Growth Rate Mean and Variance Estimators 13. Law of Large Numbers (Kelly Objectives 14. Kelly Objectives for Markowitz Portfolios 15. Central Limit Theorem Objectives 16. Optimization of Mean-Variance Objectives 17. Fortune s Formulas 18. Utility Function Objectives

3 Portfolios 12. Growth Rate Mean and Variance Estimators 1. Moment and Cumulant Generating Functions 2. Estimators from Moment Generating Functions 3. Estimators from Cumulent Generating Functions

4 Portfolios 12. Growth Rate Estimators The idea now is to treat the Markowitz portfolio associated with f as a single risky asset that can be modeled by the IID process associated with the growth rate probability density p f (X given by p f (X = q f ( e X 1 e X. The mean γ and variance θ of X are given by γ = X p f (X dx, θ = (X γ 2 p f (X dx. We know from our study of one risky asset that γ is a good proxy for reward, while θ is a good proxy for risk. Therefore we would like to estimate γ and θ in terms of the estimators ˆµ and ˆξ that we studied last time.

5 Moment and Cumulant Generating Functions. Estimators for γ and θ will be constructed from the positive function M(τ = Ex ( e τx = e τx p f (X dx. We will assume M(τ is defined for every τ in an open interval (τ mn, τ mx that contains the interval [0, 2]. It can then be shown that M(τ is infinitely differentiable over (τ mn, τ mx with M (m (τ = Ex ( X m e τx = X m e τx p f (X dx. We call M(τ the moment generating function for X because, by setting τ = 0 in the above expression, we see that the moments {Ex(X m } m=1 are generated from M(τ by the formula Ex ( X m = X m p f (X dx = M (m (0.

6 A related inifinitely differentiable function over (τ mn, τ mx is K(τ = log(m(τ = log ( Ex ( e τx. We call K(τ the cumulant generating function because the cumulants {κ m } m=1 of X are generated by the formula κ m = K (m (0. Because K (τ = Ex( X e τx Ex ( e τx, K (τ = Ex( (X K (τ 2 e τx Ex ( e τx, K (τ = Ex( (X K (τ 3 e τx Ex ( e τx, K (τ = Ex( (X K (τ 4 e τx Ex ( e τx 3K (τ 2,

7 we see that the first four cumulants of X are κ 1 = K (0 = Ex(X = γ, κ 2 = K (0 = Ex ( (X γ 2 = θ, κ 3 = K (0 = Ex ( (X γ 3, κ 4 = K (0 = Ex ( (X γ 4 3θ 2. We call these respectively the mean, variance, skewness, and kurtosis. The skewness measures asymmetry in the tails of the distribution. It is positive or negative depending on whether the fatter tail is to the right or left respectively. The kurtosis measures a relationship between the tails and the center of the distribution. It is greater for distributions with greater weight in the tails than in the center. Remark. It should be evident from the formulas on the previous slide that K (τ, K (τ, K (τ, and K (τ are the mean, variance, skewness, and kurtosis for the probability density e τx p f (X/Ex(e τx.

8 Remark. If X is normally distributed with mean γ and variance θ then p f (X = 1 2πθ exp ( (X γ2 2θ A direct calculation then shows that Ex ( e τx = 1 ( (X γ2 exp + τx dx 2πθ 2θ = 1 ( (X γ τθ2 exp + τγ + 1 2πθ 2θ 2 τ 2 θ = exp ( τγ τ 2 θ, Therefore K(τ = log(ex(e τx = τγ τ 2 θ. This shows that when X is normally distributed the skewness, kurtosis, and all other higher-order cumulants vanish. Conversely, if all these higher-order cumulants vanish then X is normally distributed.. dx

9 Remark. The cumulent generating function K(τ is strictly convex over the interval (τ mn, τ mx because K (τ > 0. Remark. We can also see that K(τ is convex over (τ mn, τ mx as follows. Let τ 0, τ 1 (τ mn, τ mx. By applying the Hölder inequality with p = 1 s 1 and p = 1 s, we see that for every s (0, 1 we have M ( (1 sτ 0 + sτ 1 = ( e (1 sτ 0X e sτ 1X p f (X dx e τ 0X p f (X dx = M(τ 0 1 s M(τ 1 s. 1 s ( By taking the logarithm of this inequality we obtain e τ 1X p f (X dx K((1 sτ 0 + sτ 1 (1 sk(τ 0 + sk(τ 1 for every s (0, 1. Therefore K(τ is a convex function over (τ mn, τ mx. s

10 Estimators from Moment Generating Functions. We will now construct estimators for γ and θ by using the moment generating function M(τ = Ex ( e τx. Because R = e X 1 and Ex(e X = M(1, we have µ = Ex(R = M(1 1. Because R µ = e X M(1 and Ex(e 2X = M(2, we have ξ = Ex ( (R µ 2 = ( M(2 M(1 2. These equations can be solved for M(1 and M(2 as M(1 = 1 + µ, M(2 = (1 + µ 2 + ξ. Therefore knowing µ and ξ is equivalent to knowing M(1 and M(2.

11 Because Ex(X = M (0 and Ex(X 2 = M (0, we see that γ = Ex(X = M (0, θ = Ex ( (X γ 2 = Ex ( X 2 γ 2 = M (0 M (0 2. Because M(0 = 1, we construct an estimator of M(τ by interpolating the values M(0, M(1, and M(2 with a quadratic polynomial as ˆM(τ = 1 + τ ( M(1 1 + τ(τ 1 1 ( 2 M(2 2M(1 + 1 = 1 + τµ τ(τ 1 ( µ 2 + ξ. By direct calculation we see that ˆM (0 = µ 1 2 (µ2 + ξ, ˆM (0 = µ 2 + ξ. (1 We then construct estimators ˆγ and ˆθ as functions of µ and ξ by ˆγ = ˆM (0 = µ 2 1 (µ2 + ξ, ˆθ = ˆM (0 ˆM (0 2 = µ 2 + ξ ( µ 2 1 (µ2 + ξ 2.

12 By replacing the µ and ξ that appear in the foregoing estimators with the estimators ˆµ = µ rf (1 1 T f + m T f, ˆξ = 1 1 w ft Vf. (2a we obtain the estimators ˆγ = ˆµ 2 1 (ˆµ 2 + ˆξ, ˆθ = ˆµ 2 + ˆξ (ˆµ 1 2 (ˆµ 2 + ˆξ 2, (2b The variance θ is generally positive, but the estimator ˆθ given above is not intrinsically positive.

13 Expanding the above expression for ˆθ in powers of ˆµ and ˆξ yields ˆθ = ˆξ + ˆµ (ˆµ 2 + ˆξ 1 4 (ˆµ 2 + ˆξ 2. The only term in this expansion that is intrinsically positive is the first one. Therefore we make the smallness assumptions ˆµ 1, ˆξ 1, and keep only through quadratic statistics i.e. through quadratic in ˆµ and linear in ˆξ. We thereby arrive at the quadratic estimators where ˆµ and ˆξ are given by (2a. ˆγ = ˆµ 1 2 (ˆµ 2 + ˆξ, ˆθ = ˆξ, (3 Remark. These smallness assumptions are very easy to check.

14 Remark. The estimators ˆγ and ˆθ given above have at least three potential sources of error: the estimators ˆM (0 and ˆM (0 as functions of µ and ξ given by (1, the estimators ˆµ and ˆξ used in (2 to approximate µ and ξ, the smallness assumptions that lead to (3. The derivation of the first estimators assumes that the returns for each Markowitz portfolio are described by a density q f (R that is narrow enough for some moment beyond the second to exist. All of these approximations should be examined carefully, especially when markets are highly volatile.

15 Estimators from Cumulent Generating Functions. We will now give an alternative derivation of estimators that uses the cumulent generating function K(τ = log(m(τ and is based on the fact that γ = K (0 and θ = K (0. It begins by observing that K(1 = log ( M(1 = log(1 + µ, K(2 = log ( M(2 = log ( (1 + µ 2 + ξ. Therefore knowing µ and ξ is equivalent to knowing K(1 and K(2. Because K(0 = 0, we construct an estimator of K(τ by interpolating the values K(0, K(1, and K(2 with a quadratic polynomial as ˆK(τ = τk(1 + τ(τ ( K(2 2K(1 = τ log(1 + µ + τ(τ 1 2 (1 1 log ξ + (1 + µ 2.

16 This yields the estimators ˆγ = ˆK (0 = log(1 + µ 1 2 log (1 + ˆθ = ˆK (0 = log ( 1 + ξ (1 + µ 2. ξ (1 + µ 2, (4 By replacing the µ and ξ that appear above with the estimators ˆµ and ˆξ given by (2a, we obtain the new estimators ˆγ = log(1 + ˆµ 2 (1 1 log ˆξ + (1 + ˆµ 2, ( (5 ˆξ ˆθ = log 1 + (1 + ˆµ 2. So long as 1 + ˆµ > 0 these estimators will be well defined and ˆθ will be positive.

17 If 1 + ˆµ > 0 and we make the smallness assumption then we obtain the estimators ˆξ (1 + ˆµ 2 1, ˆγ = log(1 + ˆµ 2 1 ˆξ (1 + ˆµ 2, ˆθ = Finally, if we make the additional smallness assumption use the fact ˆµ 1, log(1 + ˆµ = ˆµ 1 2ˆµ ˆµ3 +, ˆξ (1 + ˆµ 2. (6 and keep only through quadratic statistics then we obtain the quadratic estimators derived earlier in (3

18 Remark. The fact that both derivations lead to the same estimators gives us greater confidence in the validity the quadratic estimators. Remark. If the Markowitz portfolio specified by f has growth rates X that are normally distributed with mean γ and variance θ then we have seen that K(τ = τγ τ 2 θ. In this case we have ˆK(τ = K(τ, so the estimators ˆγ = ˆK (0 and ˆθ = ˆK (0 are exact. Remark. The biggest uncertainty associated with these estimators for ˆγ and ˆθ is usually the uncertainty inherited from the estimators for ˆµ and ˆξ.

19 Exercise. When the quadratic estimators ˆγ and ˆθ are applied to a single risky asset, they reduce to ˆγ = ˆµ 1 2(ˆµ 2 + ˆξ, ˆθ = ˆξ. Use these to estimate γ and θ for each of the following assets given the share price history {s(d} D d=0. How do these ˆγ and ˆθ compare with the unbiased estimators for γ and θ that you obtained in the previous problem? (a Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009; (b Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007; (c S&P 500 and Russell 1000 and 2000 index funds in 2009; (d S&P 500 and Russell 1000 and 2000 index funds in Exercise. Compute ˆγ and ˆθ based on daily data for the Markowitz portfolio with value equally distributed among the assets in each of the groups given in the previous exercise.

Modeling Portfolios that Contain Risky Assets Stochastic Models I: One Risky Asset

Modeling Portfolios that Contain Risky Assets Stochastic Models I: One Risky Asset C. David Levermore University of Maryland, College Park Math 420: Mathematical Modeling March 25, 2014 version c 2014